Generalized Sasakian space forms and conformal changes of metric

Generalized Sasakian space forms and conformal changes of metric Pablo Alegre and Alfonso Carriazo Dedicated to Professor H. Wefelscheid Abstract. The study of generalized Sasakian space forms is continued in this paper. The behavior of such spaces under generalized D-conformal deformations is analyzed. As a consequence, new examples of generalized Sasakian space forms are given. Mathematics Subject Classification (2010). Primary 53C25; Secondary 53D15. Keywords. Almost contact metric manifold, Sasakian manifold, Sasakian space form, D-conformal deformation, warped product, α-Sasakian manifold, β-Kenmotsu manifold, (α, β) trans-Sasakian manifold. 1. Introduction In [1], jointly with D. E. Blair, we initiated the study of generalized Sasakian space forms. There, we defined them and presented some examples, and in [2] we studied the structures of such manifolds. In our search for interesting examples (i.e., with non-constant functions) we have used different transformations; now we study D-conformal deformations of metric. Our purpose is to apply them to trans-Sasakian manifolds. The paper is organized as follows. Section 2 contains basic results about trans-Sasakian manifolds and generalized Sasakian space forms. In Section 3 we obtain trans-Sasakian manifolds by making D-conformal deformations. In the last section, the curvature tensor of a generalized Sasakian space form after a D-conformal deformation is studied. We obtain certain conditions for This paper was finished while the second author was visiting TU Berlin in June 2010. He wants to express his deepest gratitude to the Foundation Dierks von Zweck for the financial support and to Prof. Udo Simon for the warm hospitality received during that visit. Both authors are partially supported by the PAI group FQM-327 (Junta de Andaluc´ıa, Spain, 2010). The second author is also partially supported by the MEC-FEDER grant MTM2007-61248. 2 Pablo Alegre and Alfonso Carriazo such a manifold to be a new generalized Sasakian space form, and give some examples. 2. Preliminaries In this section, we recall some basic formulas about almost contact metric manifolds. For more background on contact Riemannian geometry, we recommend [3]. An odd-dimensional Riemannian manifold (M, g) is called almost contact metric manifold if there exist a (1,1) tensor field φ, a vector field ξand a 1-form ηsuch that φ2X=−X+η(X)ξ, η(ξ) = 1, g(φX, φY ) = g(X, Y )−η(X)η(Y) and g(X, ξ) = η(X), for any vector fields X, Y on M. As a consequence, φξ = 0 and η◦φ= 0. In this paper, we will say that a differentiable function fon Mdepends only on the direction of ξif X(f) = 0, for any vector field Xorthogonal to ξ. An almost contact metric manifold is said to be an (α, β)trans-Sasakian manifold, [7], if (∇Xφ)Y=α(g(X, Y )ξ−η(Y)X) + β(g(φX, Y )ξ−η(Y)φX),(2.1) which implies ∇Xξ=−αφX +β(X−η(X)ξ),(2.2) for certain differentiable functions αand β, and any X, Y on M. Those manifolds include Sasakian (α= 1, β = 0), Kenmotsu (α= 0, β = 1), cosymplectic (α= 0, β = 0), α-Sasakian (β= 0) and β-Kenmotsu (α= 0) manifolds. Of course, cosymplectic manifolds can be seen as both 0-Sasakian and 0-Kenmotsu ones. In [4], U. C. De and M. M. Tripathi proved that for an (α, β) trans- Sasakian manifold the following relation between αand βis satisfied: ξ(α)+2αβ = 0.(2.3) Moreover, J. C. Marrero said in [5] that, for dimensions greater than or equal to 5, every (α, β) trans-Sasakian manifold must be either an α-Sasakian or a β-Kenmotsu manifold, that is either αor βmust be 0. Given an almost contact metric manifold (M, φ, ξ, η, g), a φ-section of Mat p∈Mis a section Π ⊆TpMspanned by a unit vector field Xp orthogonal to ξp, and φXp. The φ-sectional curvature is defined by K(X∧ φX) = R(X, φX, φX, X). A Sasakian manifold with constant φ-sectional curvature cis called a Sasakian space form, and it is denoted by M(c). In such a case, its Riemannian curvature tensor is given by: R(X, Y )Z=c+ 3 4R1(X, Y )Z+c−1 4R2(X, Y )Z+c−1 4R3(X, Y )Z, (2.4) for any vector fields X, Y, Z in M, where R1(X, Y )Z=g(Y, Z)X−g(X, Z)Y, Generalized Sasakian space forms and conformal changes of metric 3 R2(X, Y )Z=g(X, φZ)φY −g(Y, φZ)φX + 2g(X, φY )φZ, R3(X, Y )Z=η(X)η(Z)Y−η(Y)η(Z)X+g(X, Z)η(Y)ξ−g(Y, Z)η(X)ξ. In [1] the authors, jointly with D. E. Blair, introduced the notion of generalized Sasakian space form as an almost contact metric manifold (M, φ, ξ, η, g) whose Riemannian curvature tensor satisfies R(X, Y )Z=f1R1(X, Y )Z+f2R2(X, Y )Z+f3R3(X, Y )Z, (2.5) for certain differentiable functions f1, f2, f3on M. It was denoted by M(f1, f2, f3). 3. Trans-Sasakian manifolds and generalized Sasakian space forms In [2] we gave some results for generalized Sasakian space forms doted with an α-Sasakian or a β-Kenmotsu structure. Now we continue those results. Proposition 3.1. Let M(f1, f2, f3)be an (α, β)trans-Sasakian generalized Sasakian space form. Then, the following equation holds: f1−f3+ξ(β) + β2−α2= 0.(3.1) Proof. On the one hand, as Mis an (α, β) trans-Sasakian manifold, ∇Xξis given by (2.2), so for Xorthogonal to ξ, R(X, ξ)ξ=ξ(α)φX+α(∇ξφ)X−ξ(β)X+αφ(−αφX+βX)−β(−αφX+βX), and using (2.1) and (2.3) it is R(X, ξ)ξ= (α2−β2−ξ(β))X. (3.2) On the other hand, in a generalized Sasakian space form, R(X, ξ)ξ= (f1−f3)X. (3.3) So comparing (3.2) and (3.3), and multiplying by Xwe get (3.1). ¤ Let us study some more results about αand β. Proposition 3.2. Let M(f1, f2, f3)be an (α, β)trans-Sasakian generalized Sasakian space form. Then, X(α)−φX(β) = 0 for every tangent vector field Xorthogonal to ξ. Proof. Let Xbe a vector field orthogonal to ξ. As Mis (α, β) trans-Sasakian, from (2.1) and (2.2) we obtain R(X, φX)ξ= (X(α)−φX(β))X+ (X(β) + φX(α))φX. (3.4) However, for a generalized Sasakian space form we have R(X, φX)ξ= 0. So, multiplying (3.4) by Xwe get the result. ¤ The last two propositions generalize Propositions 4.1 and 4.3 from [2]. 4 Pablo Alegre and Alfonso Carriazo 4. Trans-Sasakian manifolds and D-conformal deformations Given an almost contact metric manifold (M, φ, ξ, η, g), let us consider the generalized D-conformal deformation given by φ∗=φ, ξ∗=1 aξ, η∗=aη, g∗=bg + (a2−b)η⊗η, (4.1) where a, b are two positive functions on M. Then, it is easy to see that (M, φ∗, ξ∗, η∗, g∗) is also an almost contact metric manifold. Let us notice that we can obtain conformal, D-homothetic (see [9]) or D-conformal (in the sense of [8]) deformations, by putting a2=b,a=b= constant or a=bin (4.1), respectively. In [6], Z. Olszack considered aand bconstants, a6= 0, b > 0 but not necesarily equal, and he also called the resulting transformation a D-homothetic deformation. Moreover, let us suppose that Mis an (α, β) trans-Sasakian manifold and that a, b depend only on the direction of ξ. Therefore, we can calculate ∇∗from ∇and g∗. It is a direct calculation using Koszul’s formula and equation (2.2). It follows that the Riemannian connection ∇∗of g∗is given by: ∇∗ XY=∇XY+2(a2−b)β−ξ(b) 2a2g(φX, φY )ξ+ +ξ(b) 2b(η(Y)X+η(X)Y−2η(X)η(Y)ξ) + ξ(a2) 2a2η(X)η(Y)ξ− −2α(a2−b) 2b(η(Y)φX +η(X)φY ), (4.2) for any vector fields X, Y on M. Lemma 4.1. Let (M, φ, ξ, η, g)be an (α, β)trans-Sasakian manifold and consider a generalized D-conformal deformation on M, with a, b positive functions depending only on the direction of ξ. Then (M, φ∗, ξ∗, η∗, g∗)is also a trans-Sasakian manifold with functions: α∗=αa b, β∗=β a+ξ(b) 2ab .(4.3) Proof. On the above conditions, and using (4.2): (∇∗ Xφ)Y= (∇Xφ)Y−2(a2−b)β−ξ(b) 2a2g(φX, Y )ξ− −ξ(b) 2bη(Y)φX −2α(a2−b) 2bη(Y)X+2α(a2−b) 2bη(X)η(Y)ξ= =α(g(X, Y )ξ−η(Y)X) + β(g(φX, Y )ξ−η(Y)φX)− −2(a2−b)β−ξ(b) 2a2g(φX, Y )ξ− −ξ(b) 2bη(Y)φX −2α(a2−b) 2bη(Y)X+2α(a2−b) 2bη(X)η(Y)ξ, Generalized Sasakian space forms and conformal changes of metric 5 since Mis (α, β) trans-Sasakian. Changing to φ∗, ξ∗, η∗and g∗we obtain (∇∗ Xφ)Y=αa b(g∗(X, Y )ξ∗−η∗(Y)X)+ +µβ a+ξ(b) 2ab ¶(g∗(φ∗X, Y )ξ∗−η∗(Y)φ∗X), and so (M, φ∗, ξ∗, η∗, g∗) is a trans-Sasakian manifold with the functions given in (4.3). ¤ From this theorem we deduce that, after such a D-conformal deformation, a β-Kenmotsu manifold gives a µ0,β a+ξ(b) 2ab ¶trans-Sasakian manifold and an α-Sasakian manifold produces an µαa b,ξ(b) 2ab ¶trans-Sasakian manifold. By using this last case, it seems that we could produce examples of trans-Sasakian manifolds in any dimension with non-vanishing functions, simply by starting with an α-Sasakian manifold with α6= 0 at any point (for example, a Sasakian manifold, where α= 1), which would conflict with the theorem of Marrero mentioned above. But this is not true. Actually, under our conditions (α6= 0 and a, b depending only on ξ), both aand bare constants and what we obtain is really an ³αa b,0´trans-Sasakian manifold. We deduce this fact from the following general lemma: Lemma 4.2. Let (M, φ, ξ, η, g)be an (α, β)trans-Sasakian manifold with α6= 0at any point. If fis a function on Mdepending only on the direction of ξ, then it is constant. Proof. It is well-known that, in an (α, β) trans-Sasakian manifold, dη = αΦ. Since α6= 0 at any point, we can use Darboux’s theorem for contact manifolds, which states that, for each p∈M, there exist coordinates (t, q1, ..., qn, p1, ..., pn) such that η=dt − m X i=1 pidqi, ξ =∂ ∂t. So ½Xi=∂ ∂qi+pi ∂ ∂t, Yi=∂ ∂pi¾is a local frame of the contact distribution. Now, given fa function depending only on ξ,X(f) = 0 for all Xorthogonal to ξ. Then Xi(f) = Yi(f) = 0 and ξ(f) = [Yi, Xi](f) = Yi(Xi(f)) −Xi(Yi(f)) = 0. Therefore fis constant. ¤ We thank professor J. C. Marrero for his comments on this point. Moreover, if αis constant, then the only possible (α, β) trans-Sasakian manifolds are the α-Sasakian and β-Kenmotsu ones because, in such a case, from equation (2.3) we obtain that either αor βmust be zero. 6 Pablo Alegre and Alfonso Carriazo 5. Generalized Sasakian space forms and D-conformal deformations If, in addition, M(f1, f2, f3) is a generalized Sasakian space form with an (α, β) trans-Sasakian structure, with αand βdepending only on the direction of ξ, then, a straightforward computation using (2.1), (2.2) and (4.2) gives the curvature tensor R∗of (M, φ∗, ξ∗, η∗, g∗): R∗(X, Y )Z= =1 b·f1+β2−1 4a2b(2bβ +ξ(b))2¸R∗ 1(X, Y )Z +·f2 b−α2(a2−b) b2¸R∗ 2(X, Y )Z +·1 a2µf3+a2−b a2b(bβ2−βξ(b)) −ξ(a) aβ−ξ(b)2 4a2b−ξ(a)ξ(b) 2ab +α2 b2(b2−a4)+ +ξ2(b) 2b−ξ(b)2 2b2+ξ(b)2 4b2¶+a2−b a2bµf1+β2−1 4a2b(2bβ +ξ(b))2¶¸ {η∗(X)η∗(Z)Y−η∗(Y)η∗(Z)X} +1 b·f3−a2−b bα2−a2−b a2ξ(β)−bβ a3ξ(a) + ξ2(b) 2a2−1 2a3ξ(a)ξ(b)−ξ(b)2 2a2b¸ {g∗(X, Z)η∗(Y)ξ∗−g∗(Y, Z)η∗(X)ξ∗} + α b·ξ(b) 2b−ξ(a) a¸{g∗(Y, φZ))η∗(X)ξ∗−g∗(X, φZ)η∗(Y)ξ∗+ + 2g∗(φX, Y )η∗(Z)ξ∗} −αξ(b) 2b2{g∗(X, Z)φY −g∗(Y, Z)φX} +1 a2·2(a2−b)αβ b−α 2bξ(b) + αa bξ(a) + a2−b bξ(α)−(a2−b)αξ(b) 2b2¸ {η∗(Y)η∗(Z)φX −η∗(X)η∗(Z)φY } +αξ(b) 2b2{g∗(Z, φY )X−g∗(Z, φX)Y+ 2g∗(X, φY )Z}. (5.1) Let us note that the third and fourth addends are part of R∗ 3(X, Y )Z. Just making equal their coefficients, it holds that they can be joined if and only if f1−f3+ξ(β) + β2−α2= 0, equation that was proved in Proposition 3.1. Also, the sixth and eighth terms have the same coefficient, apart from the sign. The coefficient of the fifth addend is α b·ξ(b) 2b−ξ(a) a¸=αa2 2b2ξµb a2¶, which, using (2.3) we can write as the one of the seventh addend, apart from the sign again. Joining the appropriate terms, the curvature appears as: Generalized Sasakian space forms and conformal changes of metric 7 R∗(X, Y )Z= =1 b·f1+β2−1 4a2b(2bβ +ξ(b))2¸R∗ 1(X, Y )Z+ +1 b·f2−α2(a2−b) b¸R∗ 2(X, Y )Z+ +1 b·f3−a2−b bα2−a2−b a2ξ(β)−bβ a3ξ(a)+ +ξ2(b) 2a2−ξ(a)ξ(b) 2a3−ξ(b)2 2a2b¸R∗ 3(X, Y )Z+ +αa2 2b2ξµb a2¶{g∗(Y, φ∗Z)η∗(X)ξ∗−g∗(X, φ∗Z)η∗(Y)ξ∗+ + 2g∗(φ∗X, Y )η∗(Z)ξ∗− −η∗(Y)η∗(Z)φ∗X+η∗(X)η∗(Z)φ∗Y}+ +αξ(b) 2b2{−g∗(X, Z)φ∗Y+g∗(Y, Z)φ∗X+ +g∗(Z, φ∗Y)X−g∗(Z, φ∗X)Y+ 2g∗(X, φ∗Y)Z}. (5.2) Because of the last two terms we can state that even now, in general, a generalized D-conformal deformation of a generalized Sasakian space form is not another generalized Sasakian space form. But in dimensions greater than of equal to 5 only α-Sasakian and β-Kenmotsu manifolds exist and if a and bare constant, this equation is quite more simple. That is, under certain conditions (either on the structure or on the deformation) these terms vanish. We distinguish two cases: α6= 0 and α= 0. On the one hand, if α6= 0 it follows from Lemma 4.2 that, as α, β, a and bonly depend on ξ, they are constant. Therefore we are dealing with a D-homothetic deformation: Theorem 5.1. Let M(f1, f2, f3)be an (α, β)trans-Sasakian generalized Sasakian space form with α, β depending only on the direction of ξ. If we consider aD-homothetic deformation (in the sense of [6]), then we obtain a generalized Sasakian space form M(f∗ 1, f∗ 2, f∗ 3), with functions f∗ 1=1 b·f1+β2−bβ2 a2¸, f∗ 2=1 b·f2−α2(a2−b) b¸, f∗ 3=1 b·f3−α2(a2−b) b¸, which is endowed with an µαa b,β a¶trans-Sasakian structure. Proof. The values of f∗ 1, f∗ 2and f∗ 3are obtained from (5.2) and the fact that aand bare constants. The resultant trans-Sasakian structure is deduced from Lemma 4.1. ¤ 8 Pablo Alegre and Alfonso Carriazo Theorem 5.1 generalizes Theorem 5.5 from [1] that was given for certain β-Kenmotsu manifolds. On the other hand, if α= 0 we have: Theorem 5.2. Let M(f1, f2, f3)be a β-Kenmotsu generalized Sasakian space form. If we consider a generalized D-conformal deformation with a, b depending only on the direction of ξ, then we obtain a generalized Sasakian space form M(f∗ 1, f∗ 2, f∗ 3), with functions f∗ 1=1 b·f1+β2−1 4a2b(2bβ +ξ(b))2¸, f∗ 2=f2 b, f∗ 3=1 b·f3−a2−b a2ξ(β)−bβ a3ξ(a) + ξ2(b) 2a2−ξ(a)ξ(b) 2a3−ξ(b)2 2a2b¸, which is endowed with a µ0,β a+ξ(b) 2ab ¶trans-Sasakian structure. Proof. The values of f∗ 1, f∗ 2and f∗ 3are obtained by replacing α= 0 in (5.2). The resultant trans-Sasakian structure is deduced from Lemma 4.1. ¤ As we commented before, these D-conformal deformations englobe conformal and D-conformal (in the sense of [8]) deformations, so Theorem 5.2 generalizes Theorem 5.3, Theorem 5.4 and Theorem 5.6 from [1]. We can obtain many examples satisfying all conditions stated in Theorem 5.2. For instance, it was proved in [1] that the warped product R×fN(c) is a generalized Sasakian space form with functions f1=c−4f02 4f2, f2=c 4f2, f3=c−4f02 4f2+f00 f, where N(c) is a complex space form. Hence, if we consider Cn,CPn(4) or CHn(−4) as N(c), we obtain some specific examples in the above conditions. References [1] P. Alegre, D. E. Blair and A. Carriazo, Generalized Sasakian-space-forms. Israel J. Math. 141 (2004), 157–183. [2] P. Alegre and A. Carriazo, Structures on generalized Sasakian-space-forms. Differential Geom. Appl. 26 (2008), 656–666. [3] D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds. Birkh¨auser, 2002. [4] U. C. De and M. M. Tripathi, Ricci tensor in 3-dimensional Trans-Sasakian manifolds. Kyungpook Math. J. 43 (2003), 247–255. [5] J. C. Marrero, The local structure of trans-Sasakian manifolds. Ann. Mat. Pura Appl. 162 (1992), 77–86. [6] Z. Olszack, Curvature properties of quasi-Sasakian manifolds. Tensor, N.S. 38 (1982), 19–28. Generalized Sasakian space forms and conformal changes of metric 9 [7] J. A. Oubi˜na, New classes of almost contact metric structures. Publ. Math. Debrecen 32 (1985), 187–193. [8] S. Suguri and S. Nakayama, D-conformal deformations on almost contact metric structure. Tensor, N.S. 28 (1974), 125–129. [9] S. Tanno, The topology of contact Riemannian manifolds. Illinois J. Math. 12 (1968), 700–717. Pablo Alegre Department of Economics, Quantitative Methods and Economic History. Statistics and Operations Research Area. University Pablo de Olavide. Ctra. de Utrera, km. 1 41013 - Sevilla Spain e-mail: [email protected] Alfonso Carriazo Department of Geometry and Topology. Faculty of Mathematics. University of Sevilla. Apdo. Correos 1160 41080 – Sevilla Spain e-mail: [email protected]